Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#1875 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#121 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1155 conjugacy classes of subgroups; only the 287 normal subgroups are shown.
This group has 1155 conjugacy classes of subgroups — too many to draw. Only its 287 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(4,6)(7,8)(9,10)(11,12,16,17,15,13,14,18);(3,4,5,6)(11,13,15,12)(14,18,16,17);(8,10)(11,14,15,16)(12,17,13,18)76 conjugacy classes — view the full character table.
| maximal subgroup of | #107888, #111205, #112314, … |
| maximal normal subgroup of | #107888, #111205, #112314, … |
| maximal quotient of | C2×#8940, C2×#8943, #193449, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13895);